mirror of
https://github.com/wassname/simpeg.git
synced 2026-06-29 18:27:27 +08:00
Lindsey
634 lines
20 KiB
Python
634 lines
20 KiB
Python
from SimPEG import Survey, Problem, Utils, np, sp, Solver as SimpegSolver
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from scipy.constants import mu_0
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from SurveyFDEM import SurveyFDEM, FieldsFDEM
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from simpegEM.Base import BaseEMProblem
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from simpegEM.Utils.EMUtils import omega
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# class FieldsTDEM_e_from_b(FieldsFDEM):
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# """Fancy Field Storage for a TDEM survey."""
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# knownFields = {'b_sec': 'F'}
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# aliasFields = {
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# 'b': ['b_sec','F','b_from_bsec'],
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# 'e': ['b','E','e_from_b']
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# }
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# def startup(self):
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# self.MeSigmaI = self.survey.prob.MeSigmaI
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# self.edgeCurlT = self.survey.prob.mesh.edgeCurl.T
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# self.MfMui = self.survey.prob.MfMui
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# def e_from_b(self, b, txInd, timeInd):
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# # TODO: implement non-zero js
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# return self.MeSigmaI*(self.edgeCurlT*(self.MfMui*b))
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# def e_from_bDeriv(self, b, txInd, timeInd):
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# # TODO: implement non-zero js
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# return self.MeSigmaI*(self.edgeCurlT*(self.MfMui*b))
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# def calcFields(self, sol, freq, fieldType, adjoint=False):
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# e = sol
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# if fieldType == 'e':
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# return e
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# elif fieldType == 'b':
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# if not adjoint:
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# b = - self.mesh.edgeCurl * e
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# b = 1./(1j*omega(freq)) * b
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# else:
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# b = -(1./(1j*omega(freq))) * ( self.mesh.edgeCurl.T * e )
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# return b
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# raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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# def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
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# e = sol
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# if fieldType == 'e':
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# return None
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# elif fieldType == 'b':
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# return None
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# raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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class BaseFDEMProblem(BaseEMProblem):
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"""
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We start by looking at Maxwell's equations in the electric field \\(\\vec{E}\\) and the magnetic flux density \\(\\vec{B}\\):
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.. math::
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\\nabla \\times \\vec{E} + i \\omega \\vec{B} = 0 \\\\
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\\nabla \\times \\mu^{-1} \\vec{B} - \\sigma \\vec{E} = \\vec{J_s}
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"""
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surveyPair = SurveyFDEM
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# fieldsPair = FieldsFDEM
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def forward(self, m, RHS, CalcFields):
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# F = self.fieldsPair(self.mesh, self.survey)
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F = FieldsFDEM(self.mesh, self.survey)
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for freq in self.survey.freqs:
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A = self.getA(freq)
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rhs = RHS(freq)
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Ainv = self.Solver(A, **self.solverOpts)
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sol = Ainv * rhs
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for fieldType in self.storeTheseFields:
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Txs = self.survey.getTransmitters(freq)
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F[Txs, fieldType] = CalcFields(sol, freq, fieldType)
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# Txs = self.survey.getTransmitters(freq)
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# F[Txs, 'e_sec'] = sol
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return F
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def Jvec(self, m, v, u=None):
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if u is None:
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u = self.fields(m)
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self.curModel = m
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Jv = self.dataPair(self.survey)
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for freq in self.survey.freqs:
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A = self.getA(freq)
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Ainv = self.Solver(A, **self.solverOpts)
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for tx in self.survey.getTransmitters(freq):
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u_tx = u[tx, self.solType]
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w = self.getADeriv(freq, u_tx, v)
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Ainvw = Ainv * w
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for rx in tx.rxList:
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fAinvw = self.calcFields(Ainvw, freq, rx.projField)
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P = lambda v: rx.projectFieldsDeriv(tx, self.mesh, u, v)
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Jv[tx, rx] = - P(fAinvw)
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df_dm = self.calcFieldsDeriv(u_tx, freq, rx.projField, v)
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if df_dm is not None:
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Jv[tx, rx] += P(df_dm)
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return Utils.mkvc(Jv)
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def Jtvec(self, m, v, u=None):
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if u is None:
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u = self.fields(m)
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self.curModel = m
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# Ensure v is a data object.
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if not isinstance(v, self.dataPair):
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v = self.dataPair(self.survey, v)
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Jtv = np.zeros(self.mapping.nP)
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for freq in self.survey.freqs:
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AT = self.getA(freq).T
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ATinv = self.Solver(AT, **self.solverOpts)
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for tx in self.survey.getTransmitters(freq):
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u_tx = u[tx, self.solType]
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for rx in tx.rxList:
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PTv = rx.projectFieldsDeriv(tx, self.mesh, u, v[tx, rx], adjoint=True)
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fPTv = self.calcFields(PTv, freq, rx.projField, adjoint=True)
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w = ATinv * fPTv
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Jtv_rx = - self.getADeriv(freq, u_tx, w, adjoint=True)
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df_dm = self.calcFieldsDeriv(u_tx, freq, rx.projField, PTv, adjoint=True)
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if df_dm is not None:
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Jtv_rx += df_dm
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real_or_imag = rx.projComp
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if real_or_imag == 'real':
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Jtv += Jtv_rx.real
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elif real_or_imag == 'imag':
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Jtv += - Jtv_rx.real
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else:
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raise Exception('Must be real or imag')
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return Jtv
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def getSource(self, freq):
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"""
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE or nF, nTx)
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:return: RHS
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"""
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Txs = self.survey.getTransmitters(freq)
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j_m = range(len(Txs))
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j_e = range(len(Txs))
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for i, tx in enumerate(Txs):
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j_m[i], j_e[i] = tx.getSource(self)
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return j_m, j_e
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# return np.concatenate(rhs).reshape((-1, len(Txs)), order='F') #, np.concatenate(j_e).reshape((-1, len(Txs)), order='F')
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##########################################################################################
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################################ E-B Formulation #########################################
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##########################################################################################
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class ProblemFDEM_e(BaseFDEMProblem):
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"""
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By eliminating the magnetic flux density using
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.. math::
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\\vec{B} = \\frac{-1}{i\\omega}\\nabla\\times\\vec{E},
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we can write Maxwell's equations as a second order system in \\ \\vec{E} \\ only:
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.. math::
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\\nabla \\times \\mu^{-1} \\nabla \\times \\vec{E} + i \\omega \\sigma \\vec{E} = \\vec{J_s}
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This is the definition of the Forward Problem using the E-formulation of Maxwell's equations.
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"""
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solType = 'e'
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# _fieldType = 'e'
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# fieldsPair = FieldsFDEM_e
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def __init__(self, model, **kwargs):
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BaseFDEMProblem.__init__(self, model, **kwargs)
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def getA(self, freq):
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"""
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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mui = self.MfMui
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sig = self.MeSigma
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C = self.mesh.edgeCurl
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return C.T*mui*C + 1j*omega(freq)*sig
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def getADeriv(self, freq, u, v, adjoint=False):
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sig = self.curModel.transform
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dsig_dm = self.curModel.transformDeriv
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dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig)(u)
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if adjoint:
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return 1j * omega(freq) * ( dsig_dm.T * ( dMe_dsig.T * v ) )
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return 1j * omega(freq) * ( dMe_dsig * ( dsig_dm * v ) )
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def getRHS(self, freq):
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"""
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nTx)
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:return: RHS
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"""
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j_m, j_g = self.getSource(freq)
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nTx_freq = self.survey.nTxByFreq[freq]
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RHS = 1j*np.zeros([self.mesh.nE, nTx_freq])
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C = self.mesh.edgeCurl
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MfMui = self.MfMui
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for ii in range(nTx_freq):
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if j_m[ii] is not None:
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RHS[:, ii] += C.T * (MfMui * j_m[ii])
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if j_g[ii] is not None:
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RHS[:, ii] += -1j*omega(freq)*j_g[ii]
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return RHS
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def calcFields(self, sol, freq, fieldType, adjoint=False):
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e = sol
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if fieldType == 'e':
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return e
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elif fieldType == 'b':
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if not adjoint:
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b = - self.mesh.edgeCurl * e
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b = 1./(1j*omega(freq)) * b
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else:
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b = -(1./(1j*omega(freq))) * ( self.mesh.edgeCurl.T * e )
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return b
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
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e = sol
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if fieldType == 'e':
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return None
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elif fieldType == 'b':
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return None
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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class ProblemFDEM_b(BaseFDEMProblem):
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"""
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Solving for b!
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"""
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solType = 'b'
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def __init__(self, model, **kwargs):
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BaseFDEMProblem.__init__(self, model, **kwargs)
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def getA(self, freq):
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"""
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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mui = self.MfMui
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sigI = self.MeSigmaI
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C = self.mesh.edgeCurl
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iomega = 1j * omega(freq) * sp.eye(self.mesh.nF)
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A = C*sigI*C.T*mui + iomega
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if self._makeASymmetric is True:
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return mui.T*A
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return A
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def getADeriv(self, freq, u, v, adjoint=False):
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mui = self.MfMui
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C = self.mesh.edgeCurl
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sig = self.curModel.transform
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dsig_dm = self.curModel.transformDeriv
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#TODO: This only works if diagonal (no tensors)...
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dMeSigmaI_dI = - self.MeSigmaI**2
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vec = (C.T*(mui*u))
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dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig)(vec)
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if adjoint:
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if self._makeASymmetric is True:
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v = mui * v
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return dsig_dm.T * ( dMe_dsig.T * ( dMeSigmaI_dI.T * ( C.T * v ) ) )
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if self._makeASymmetric is True:
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return mui.T * ( C * ( dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) ) ) )
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return C * ( dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) ) )
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def getRHS(self, freq):
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"""
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nTx)
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:return: RHS
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"""
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j_m, j_g = self.getSource(freq)
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nTx_freq = self.survey.nTxByFreq[freq]
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RHS = 1j*np.zeros([self.mesh.nF, nTx_freq])
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C = self.mesh.edgeCurl
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MfSigmai = self.MfSigmai
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for ii in range(nTx_freq):
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if j_m[ii] is not None:
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RHS[:,ii] += j_m[ii]
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if j_g[ii] is not None:
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RHS[:,ii] += C * ( MfSigmai * j_g[ii] )
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if self._makeASymmetric is True:
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mui = self.MfMui
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return mui.T*RHS
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return RHS
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def calcFields(self, sol, freq, fieldType, adjoint=False):
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b = sol
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if fieldType == 'e':
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if not adjoint:
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e = self.MeSigmaI * ( self.mesh.edgeCurl.T * ( self.MfMui * b ) )
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else:
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e = self.MfMui.T * ( self.mesh.edgeCurl * ( self.MeSigmaI.T * b ) )
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return e
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elif fieldType == 'b':
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return b
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
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b = sol
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if fieldType == 'e':
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sig = self.curModel.transform
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dsig_dm = self.curModel.transformDeriv
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C = self.mesh.edgeCurl
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mui = self.MfMui
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#TODO: This only works if diagonal (no tensors)...
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dMeSigmaI_dI = - self.MeSigmaI**2
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vec = C.T * ( mui * b )
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dMe_dsig = self.mesh.getEdgeInnerProductDeriv(sig)(vec)
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if not adjoint:
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return dMeSigmaI_dI * ( dMe_dsig * ( dsig_dm * v ) )
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else:
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return dsig_dm.T * ( dMe_dsig.T * ( dMeSigmaI_dI.T * v ) )
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elif fieldType == 'b':
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return None
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raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
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##########################################################################################
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################################ H-J Formulation #########################################
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##########################################################################################
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class ProblemFDEM_j(BaseFDEMProblem):
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"""
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Using the H-J formulation of Maxwell's equations
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.. math::
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\\nabla \\times \\sigma^{-1} \\vec{J} + i\\omega\\mu\\vec{H} = 0
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\\nabla \\times \\vec{H} - \\vec{J} = \\vec{J_s}
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Since \(\\vec{J}\) is a flux and \(\\vec{H}\) is a field, we discretize \(\\vec{J}\) on faces and \(\\vec{H}\) on edges.
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For this implementation, we solve for J using \( \\vec{H} = - (i\\omega\\mu)^{-1} \\nabla \\times \\sigma^{-1} \\vec{J} \) :
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.. math::
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\\nabla \\times ( \\mu^{-1} \\nabla \\times \\sigma^{-1} \\vec{J} ) + i\\omega \\vec{J} = - i\\omega\\vec{J_s}
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We discretize this to:
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.. math::
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(\\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega ) \\mathbf{j} = - i\\omega \\mathbf{j_s}
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.. note::
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This implementation does not yet work with full anisotropy!!
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"""
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solType = 'j'
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storeTheseFields = ['j','h']
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def __init__(self, model, **kwargs):
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BaseFDEMProblem.__init__(self, model, **kwargs)
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def getA(self, freq):
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"""
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Here, we form the operator \(\\mathbf{A}\) to solce
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.. math::
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\\mathbf{A} = \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} + i\\omega
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:param float freq: Frequency
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:rtype: scipy.sparse.csr_matrix
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:return: A
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"""
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MeMuI = self.MeMuI
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MfSigi = self.MfSigmai
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C = self.mesh.edgeCurl
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iomega = 1j * omega(freq) * sp.eye(self.mesh.nF)
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A = C * MeMuI * C.T * MfSigi + iomega
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if self._makeASymmetric is True:
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return MfSigi.T*A
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return A
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def getADeriv(self, freq, u, v, adjoint=False):
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"""
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In this case, we assume that electrical conductivity, \(\\sigma\) is the physical property of interest (i.e. \(\sigma\) = model.transform). Then we want
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.. math::
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\\frac{\mathbf{A(\\sigma)} \mathbf{v}}{d \\mathbf{m}} &= \\mathbf{C} \\mathbf{M^e_{mu^{-1}}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{m}}
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&= \\mathbf{C} \\mathbf{M^e_{mu}^{-1}} \\mathbf{C^T} \\frac{d \\mathbf{M^f_{\\sigma^{-1}}}}{d \\mathbf{\\sigma^{-1}}} \\frac{d \\mathbf{\\sigma^{-1}}}{d \\mathbf{\\sigma}} \\frac{d \\mathbf{\\sigma}}{d \\mathbf{m}}
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"""
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MeMuI = self.MeMuI
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MfSigi = self.MfSigmai
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C = self.mesh.edgeCurl
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sig = self.curModel.transform
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sigi = 1/sig
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dsig_dm = self.curModel.transformDeriv
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dsigi_dsig = -Utils.sdiag(sigi)**2
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dMf_dsigi = self.mesh.getFaceInnerProductDeriv(sigi)(u)
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if adjoint:
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if self._makeASymmetric is True:
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v = MfSigi * v
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return dsig_dm.T * ( dsigi_dsig.T *( dMf_dsigi.T * ( C * ( MeMuI.T * ( C.T * v ) ) ) ) )
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if self._makeASymmetric is True:
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return MfSigi.T * ( C * ( MeMuI * ( C.T * ( dMf_dsigi * ( dsigi_dsig * ( dsig_dm * v ) ) ) ) ) )
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return C * ( MeMuI * ( C.T * ( dMf_dsigi * ( dsigi_dsig * ( dsig_dm * v ) ) ) ) )
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def getRHS(self, freq):
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"""
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:param float freq: Frequency
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:rtype: numpy.ndarray (nE, nTx)
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:return: RHS
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"""
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j_m, j_g = self.getSource(freq)
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nTx_freq = self.survey.nTxByFreq[freq]
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RHS = 1j*np.zeros([self.mesh.nF, nTx_freq])
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C = self.mesh.edgeCurl
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MeMuI = self.MeMuI
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for ii in range(nTx_freq):
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if j_m[ii] is not None:
|
|
RHS[:,ii] += C * (MeMuI * j_m[ii])
|
|
|
|
if j_g[ii] is not None:
|
|
RHS[:,ii] += -1j * omega(freq) * j_g[ii]
|
|
|
|
|
|
if self._makeASymmetric is True:
|
|
MfSigi = self.MfSigmai
|
|
return MfSigi.T*RHS
|
|
|
|
return RHS
|
|
|
|
|
|
def calcFields(self, sol, freq, fieldType, adjoint=False):
|
|
j = sol
|
|
if fieldType == 'j':
|
|
return j
|
|
elif fieldType == 'h':
|
|
MeMuI = self.MeMuI
|
|
C = self.mesh.edgeCurl
|
|
MfSigi = self.MfSigmai
|
|
if not adjoint:
|
|
h = -(1./(1j*omega(freq))) * MeMuI * ( C.T * ( MfSigi * j ) )
|
|
else:
|
|
h = -(1./(1j*omega(freq))) * MfSigi.T * ( C * ( MeMuI.T * j ) )
|
|
return h
|
|
raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
|
|
|
|
def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
|
|
j = sol
|
|
if fieldType == 'j':
|
|
return None
|
|
elif fieldType == 'h':
|
|
MeMuI = self.MeMuI
|
|
C = self.mesh.edgeCurl
|
|
sig = self.curModel.transform
|
|
sigi = 1/sig
|
|
dsig_dm = self.curModel.transformDeriv
|
|
dsigi_dsig = -Utils.sdiag(sigi)**2
|
|
dMf_dsigi = self.mesh.getFaceInnerProductDeriv(sigi)(j)
|
|
sigi = self.MfSigmai
|
|
if not adjoint:
|
|
return -(1./(1j*omega(freq))) * MeMuI * ( C.T * ( dMf_dsigi * ( dsigi_dsig * ( dsig_dm * v ) ) ) )
|
|
else:
|
|
return -(1./(1j*omega(freq))) * dsig_dm.T * ( dsigi_dsig.T * ( dMf_dsigi.T * ( C * ( MeMuI.T * v ) ) ) )
|
|
raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
|
|
|
|
|
|
|
|
|
|
# Solving for h! - using primary- secondary approach
|
|
class ProblemFDEM_h(BaseFDEMProblem):
|
|
"""
|
|
Using the H-J formulation of Maxwell's equations
|
|
|
|
.. math::
|
|
\\nabla \\times \\sigma^{-1} \\vec{J} + i\\omega\\mu\\vec{H} = 0
|
|
\\nabla \\times \\vec{H} - \\vec{J} = \\vec{J_s}
|
|
|
|
Since \(\\vec{J}\) is a flux and \(\\vec{H}\) is a field, we discretize \(\\vec{J}\) on faces and \(\\vec{H}\) on edges.
|
|
|
|
For this implementation, we solve for J using \( \\vec{J} = \\nabla \\times \\vec{H} - \\vec{J_s} \)
|
|
|
|
.. math::
|
|
\\nabla \\times \\sigma^{-1} \\nabla \\times \\vec{H} + i\\omega\\mu\\vec{H} = \\nabla \\times \\sigma^{-1} \\vec{J_s}
|
|
|
|
We discretize and solve
|
|
|
|
.. math::
|
|
(\\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} \\mathbf{C} + i\\omega \\mathbf{M_{\mu}} ) \\mathbf{h} = \\mathbf{C^T} \\mathbf{M^f_{\\sigma^{-1}}} \\vec{J_s}
|
|
|
|
.. note::
|
|
This implementation does not yet work with full anisotropy!!
|
|
|
|
"""
|
|
|
|
solType = 'h'
|
|
storeTheseFields = ['j','h']
|
|
|
|
def __init__(self, model, **kwargs):
|
|
BaseFDEMProblem.__init__(self, model, **kwargs)
|
|
|
|
def getA(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: scipy.sparse.csr_matrix
|
|
:return: A
|
|
"""
|
|
|
|
MeMu = self.MeMu
|
|
MfSigi = self.MfSigmai
|
|
C = self.mesh.edgeCurl
|
|
|
|
return C.T * MfSigi * C + 1j*omega(freq)*MeMu
|
|
|
|
def getADeriv(self, freq, u, v, adjoint=False):
|
|
|
|
MeMu = self.MeMu
|
|
C = self.mesh.edgeCurl
|
|
sig = self.curModel.transform
|
|
sigi = 1/sig
|
|
dsig_dm = self.curModel.transformDeriv
|
|
dsigi_dsig = -Utils.sdiag(sigi)**2
|
|
|
|
dMf_dsigi = self.mesh.getFaceInnerProductDeriv(sigi)(C*u)
|
|
|
|
if adjoint:
|
|
return (dsig_dm.T * (dsigi_dsig.T * (dMf_dsigi.T * (C * v))))
|
|
return (C.T * (dMf_dsigi * (dsigi_dsig * (dsig_dm * v))))
|
|
|
|
|
|
def getRHS(self, freq):
|
|
"""
|
|
:param float freq: Frequency
|
|
:rtype: numpy.ndarray (nE, nTx)
|
|
:return: RHS
|
|
"""
|
|
|
|
j_m, j_g = self.getSource(freq)
|
|
nTx_freq = self.survey.nTxByFreq[freq]
|
|
RHS = 1j*np.zeros([self.mesh.nE, nTx_freq])
|
|
|
|
C = self.mesh.edgeCurl
|
|
MfSigmai = self.MfSigmai
|
|
|
|
for ii in range(nTx_freq):
|
|
if j_m[ii] is not None:
|
|
RHS[:,ii] += j_m[ii]
|
|
|
|
if j_g[ii] is not None:
|
|
RHS[:,ii] += C.T * ( MfSigmai * j_g[ii] )
|
|
|
|
return RHS
|
|
|
|
|
|
def calcFields(self, sol, freq, fieldType, adjoint=False):
|
|
h = sol
|
|
if fieldType == 'j':
|
|
C = self.mesh.edgeCurl
|
|
if adjoint:
|
|
return C.T*h
|
|
return C*h
|
|
elif fieldType == 'h':
|
|
return h
|
|
raise NotImplementedError('fieldType "%s" is not implemented.' % fieldType)
|
|
|
|
def calcFieldsDeriv(self, sol, freq, fieldType, v, adjoint=False):
|
|
return None
|